Problem: Determine how many solutions exist for the system of equations. ${-12x-2y = -14}$ ${x+y = -9}$
Answer: Convert both equations to slope-intercept form: ${-12x-2y = -14}$ $-12x{+12x} - 2y = -14{+12x}$ $-2y = -14+12x$ $y = 7-6x$ ${y = -6x+7}$ ${x+y = -9}$ $x{-x} + y = -9{-x}$ $y = -9-x$ ${y = -x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -6x+7}$ ${y = -x-9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.